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If 2^(5x)=(16)/(2^(1-x)), find x.

If $2^{5 x}=\frac{16}{2^{1-x}}$ , find $x$ .

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Powers with decimal and fractional bases

Full solution

Q. If

2^{5 x}=\frac{16}{2^{1-x}}

, find

x

.

Write Equation Given: Write down the equation given in the problem. $\newline$ We are given the equation $2^{5x} = \frac{16}{2^{1-x}}$ .
Recognize Power of $2$ : Recognize that $16$ can be written as a power of $2$ . $16$ is $2$ raised to the power of $4$ , so we can rewrite the equation as $2^{5x} = \frac{2^4}{2^{1-x}}$ .
Combine Powers of $2$ : Use the property of exponents to combine the powers of $2$ on the right side of the equation. $\newline$ When dividing like bases with exponents, we subtract the exponents: $2^{5x} = 2^{4 - (1-x)}$ .
Simplify Exponent: Simplify the exponent on the right side of the equation. $2^{5x} = 2^{4 - 1 + x}$ which simplifies to $2^{5x} = 2^{3 + x}$ .
Set Exponents Equal: Since the bases are the same and the equation is an equality, the exponents must be equal. $\newline$ Set the exponents equal to each other: $5x = 3 + x$ .
Solve for x: Solve for x. $\newline$ Subtract $x$ from both sides to get $5x - x = 3$ , which simplifies to $4x = 3$ . $\newline$ Then divide both sides by $4$ to get $x = \frac{3}{4}$ .

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